Readings

The readings below were found in the textbook: Buy at Amazon Saff, Edward B., and Arthur David Snider. Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics. 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2002. ISBN: 0139078746. The problems bold-faced below seemed especially well suited for the students' practice and for discussion at the recitations.

lec # TOPICS readings reading assignments
I. Complex Algebra and Functions
1 Algebra of Complex Numbers

Complex Plane

Polar Form		 Read 1.1 - 1.3
 p5: 8, 9,24

p12: 6, 7fq,16

p22: 6, 7eq, 8, 11, 17
2 cis(y) = exp(iy)

Powers

Geometric Series		 Read 1.4, 1.5
 p31: 1, 3, 4, 11, 12a,14

p37: 4, 5d, 9, 10, 11, 21b
3 Functions of Complex Variable

Analyticity		 Read 2.1 - 2.3 p56: 4, 5

p63: 6, 7, 11cd

p70: 4, 7c, 13, 15
4 Cauchy-Riemann Conditions

Harmonic Functions		 Read 2.4, 2.5
 p77: 1, 3, 6, 8, 11

p84: 2, 3bc, 6, 9, 12, 18
5 Simple Mappings: az+b, z2, √z

Idea of Conformality		 Skim pp. 377-79, 383-87 p57: 7, 8, 9, 11,13

p71: 8

p392: 1, 3ae
6 Complex Exponential Reread 1.4

Begin 3.2		 p32: 20a, 23a

p115: 3, 5ab, 9a, 14a, 17ab, 20
7 Complex Trigonometric and Hyperbolic Functions Finish 3.2 p115: 5cdef, 11, 14bcd, 17c, 18b, 23
8 Complex Logarithm Read 3.3 p123: 1, 4, 5, 6, 8, 11, 12, 19
9 Complex Powers

Inverse Trig. Functions		 Read 3.5 p136: 1, 4, 5, 7, 9, 11, 19
10 Broad Review ... Probably focusing on sin-1z
II. Complex Integration
11 Contour Integrals Read 4.1, 4.2 p160: 2, 4, 5, 13

p171: 3b, 6c, 9, 10, 11, 14
12 Path Independence Read 4.3

Skim 4.4a		 p178: 1aeh, 2, 5, 6, 7, 11
Exam 1
13 Cauchy's Integral Theorem Read 4.4b p200: 6, 9, 10, 13, 15, 17, 18, 20
14 Cauchy's Integral Formula

Higher Derivatives		 Read 4.5 p212: 1, 3abd, 5, 6, 8, 13
15 Bounds

Liouville's Theorem

Maximum Modulus Principle		 Read 4.6 p219: 1, 7, 10, 14, 15, 16, 18
16 Mean-value Theorems

Fundamental Theorem of Algebra		 Skim 4.7 p225: 4, 6, 7, 11 + 14?!
17 Radius of Convergence of Taylor Series Read 5.2, 5.3 p249: 1ad, 4, 5be, 11ab, 18a

p259: 7, 13a
III. Residue Calculus
18 Laurent Series Read 5.5 p276: 3, 5, 6, 7a, 10
19 Poles

Essential Singularities

Point at Infinity		 Read 5.6, 5.7 p285: 1, 2, 3, 5, 8, 12

p117: 25

p290: 1, 4, 6
20 Residue Theorem

Integrals around Unit Circle		 Read 6.1, 6.2 p313: 1adq, 3abce, 5, 7

p317: 2, 6, 9
21 Real Integrals From -∞ to +∞

Conversion to cx Contours		 Read 6.3 p325: 1, 4, 6, 10, 11, 13
22 Ditto ... Including Trig. Functions

Jordan's Lemma		 Read 6.4 p336: 1, 3, 7, 11, 12
Exam 2
23 Singularity on Path of Integration

Principal Values		 Read 6.5 p344: 1a, 3, 5, 10, 12
24 Integrals involving Multivalued Functions Read 6.6

Skim 6.7		 p354: 1, 2, 4, 8, 10, 13

p364: 7, 8, 9 (all 3 w/o Rouche!)
IV. Conformal Mapping
25 Invariance of Laplace's Equation Read 7.1 p374: 1, 3, 4
26 Conformality again

Inversion Mappings		 Read 7.2+3 to p389 p382: 1b, 3, 7*, 11, 13 (* see also p57: 6 + p85: 11)
27 Bilinear/Mobius Transformations Finish 7.3

Skim 7.4		 p392: 3cd, 6, 7b, 8, 11

p405: 18
28 Applications I Read 7.6 430: 1, 2, 3, 6, 10

p416: 4(!)
29 Applications II Read 7.7 p440: 3, 5, 6
V. Fourier Series and Transforms
30 Complex Fourier Series Read 8.1 to p453 p459: 1a, 2ac, 5, 7ac
31 Oscillating Systems

Periodic Functions		 Skim 3.6 p143: 1b, 3, 5

p393: 13(!)

p461: 9, 10
32 Questions of Convergence

Scanning Function

Gibbs Phenomenon		 Finish 8.1 p460: 3, 6, 8, 11
33 Toward Fourier Transforms Read 8.2 p473: 1abd, 2, 3abc
34 Applications of FTs Read 8.2 again p473: 6abc, 7, 8
Exam 3
35 Special Topic: The Magic of FFTs I Reread pp. 457-59 p462: 12(!)
36 Special Topic: The Magic of FFTs II
Final Exam